Falcons fly-half Toby Flood ran the game in the first half, scoring two penalties and a fine solo try.
Connacht struggled to get into the game and could only reply with a Paul Warwick penalty just before the break.
Both sides had few chances in a drab second period before Mike McCarthy made sure of the win with a close-range try.
Newcastle director of rugby Rob Andrew
"It was a shame that the elements closed in on us in the second half and we probably tried to force it a bit too much, but we played with passion and intensity, which were two of the things we talked about in the week.
"On top of that there was a lot of skill in poor conditions, and credit to Connacht because they battled to the death and didn't let us get out of sight.
"Had we not played as technically well as we did, Connacht were up for this and may have come out on top."
Newcastle: Burke; May, Noon, Mayerhofler, Elliot; Flood, Charlton; Williams, Thompson, Ward; Gross, Parling; McCarthy, Harris, Charvis, Morris.
Replacements: Walder for Elliot (75), Grindal for Charlton (47), Morris for Williams (40), Long for Thompson (40), Perry for Gross (59), Finegan for Parling (80).
Connacht: McHugh, Mostyn, Mailei, Matthews, McPhillips, Warwick, Keane, Hogan, Fogarty, Knoop, Gannon, Farley, Muldoon, Lacey, Rigney.
Replacements: Slemen for McPhillips (51), Tierney for Keane (67), Merrigan for Fogarty (80), Clarke for Knoop (70), Short for Gannon (45), Swift for Rigney (70), Williams for Slemen (54).
Ref: Eric Darriere (France).